how do I write 25a^2 - 30ab + 9b^2 as the square of a binomial
note that 25a^2 and 9b^2 are perfect squares, so you probably want
(5a-3b)^2
Now just check to be sure that produces a -30ab term in the middle.
whew, it does :)
To write the expression 25a^2 - 30ab + 9b^2 as the square of a binomial, follow these steps:
Step 1: Identify the perfect square terms.
In this case, the perfect square terms are 25a^2 and 9b^2.
Step 2: Take the square root of the perfect square terms.
The square root of 25a^2 is 5a.
The square root of 9b^2 is 3b.
Step 3: Rewrite the original expression using the square roots.
25a^2 - 30ab + 9b^2 can be rewritten as (5a - 3b)^2.
Therefore, the expression 25a^2 - 30ab + 9b^2 can be written as the square of the binomial (5a - 3b).
To write the expression 25a^2 - 30ab + 9b^2 as the square of a binomial, you can use the formula for the square of a binomial, which is (a + b)^2 = a^2 + 2ab + b^2.
Let's break down the given expression and try to match it with this formula.
25a^2 - 30ab + 9b^2
First, we look at the first term, 25a^2. It has a coefficient of 25 and the variable a raised to the power of 2.
Next, we examine the last term, 9b^2. It has a coefficient of 9 and the variable b raised to the power of 2.
Now, we focus on the middle term, -30ab. It has a coefficient of -30 and contains both a and b, but the powers are 1 each.
In order to match this expression with the square of a binomial, we take half of the coefficient of the middle term and square it. Half of -30 is -15, and (-15)^2 is 225.
Now, using the formula (a + b)^2 = a^2 + 2ab + b^2, we substitute the appropriate values:
25a^2 - 30ab + 9b^2 = (5a - 3b)^2
Therefore, the square of the binomial that is equivalent to 25a^2 - 30ab + 9b^2 is (5a - 3b)^2.